TSTP Solution File: GRP654+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP654+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 01:19:27 EDT 2023
% Result : Theorem 7.59s 1.37s
% Output : Proof 8.99s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : GRP654+1 : TPTP v8.1.2. Released v4.0.0.
% 0.10/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n011.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Mon Aug 28 22:54:11 EDT 2023
% 0.14/0.34 % CPUTime :
% 7.59/1.37 Command-line arguments: --no-flatten-goal
% 7.59/1.37
% 7.59/1.37 % SZS status Theorem
% 7.59/1.37
% 8.62/1.47 % SZS output start Proof
% 8.62/1.47 Take the following subset of the input axioms:
% 8.62/1.47 fof(f01, axiom, ![B, A]: mult(A, ld(A, B))=B).
% 8.62/1.47 fof(f02, axiom, ![B2, A2]: ld(A2, mult(A2, B2))=B2).
% 8.62/1.47 fof(f03, axiom, ![B2, A2]: mult(rd(A2, B2), B2)=A2).
% 8.62/1.47 fof(f04, axiom, ![B2, A2]: rd(mult(A2, B2), B2)=A2).
% 8.62/1.47 fof(f05, axiom, ![C, B2, A2]: mult(A2, mult(B2, mult(A2, C)))=mult(mult(mult(A2, B2), A2), C)).
% 8.62/1.47 fof(goals, conjecture, ?[X0]: ![X1]: (mult(X1, X0)=X1 & mult(X0, X1)=X1)).
% 8.62/1.47
% 8.62/1.47 Now clausify the problem and encode Horn clauses using encoding 3 of
% 8.62/1.47 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 8.62/1.47 We repeatedly replace C & s=t => u=v by the two clauses:
% 8.62/1.47 fresh(y, y, x1...xn) = u
% 8.62/1.47 C => fresh(s, t, x1...xn) = v
% 8.62/1.47 where fresh is a fresh function symbol and x1..xn are the free
% 8.62/1.47 variables of u and v.
% 8.62/1.47 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 8.62/1.47 input problem has no model of domain size 1).
% 8.62/1.47
% 8.62/1.47 The encoding turns the above axioms into the following unit equations and goals:
% 8.62/1.47
% 8.62/1.47 Axiom 1 (f02): ld(X, mult(X, Y)) = Y.
% 8.62/1.47 Axiom 2 (f04): rd(mult(X, Y), Y) = X.
% 8.62/1.47 Axiom 3 (f01): mult(X, ld(X, Y)) = Y.
% 8.62/1.47 Axiom 4 (f03): mult(rd(X, Y), Y) = X.
% 8.62/1.47 Axiom 5 (f05): mult(X, mult(Y, mult(X, Z))) = mult(mult(mult(X, Y), X), Z).
% 8.62/1.47
% 8.62/1.47 Lemma 6: rd(mult(X, mult(Y, mult(X, Z))), Z) = mult(mult(X, Y), X).
% 8.62/1.47 Proof:
% 8.62/1.47 rd(mult(X, mult(Y, mult(X, Z))), Z)
% 8.62/1.47 = { by axiom 5 (f05) }
% 8.62/1.47 rd(mult(mult(mult(X, Y), X), Z), Z)
% 8.62/1.47 = { by axiom 2 (f04) }
% 8.62/1.47 mult(mult(X, Y), X)
% 8.62/1.47
% 8.62/1.47 Lemma 7: rd(mult(X, mult(Y, Z)), ld(X, Z)) = mult(mult(X, Y), X).
% 8.62/1.47 Proof:
% 8.62/1.47 rd(mult(X, mult(Y, Z)), ld(X, Z))
% 8.62/1.48 = { by axiom 3 (f01) R->L }
% 8.62/1.48 rd(mult(X, mult(Y, mult(X, ld(X, Z)))), ld(X, Z))
% 8.62/1.48 = { by lemma 6 }
% 8.62/1.48 mult(mult(X, Y), X)
% 8.62/1.48
% 8.62/1.48 Lemma 8: mult(mult(X, rd(Y, Z)), X) = rd(mult(X, Y), ld(X, Z)).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(mult(X, rd(Y, Z)), X)
% 8.62/1.48 = { by lemma 7 R->L }
% 8.62/1.48 rd(mult(X, mult(rd(Y, Z), Z)), ld(X, Z))
% 8.62/1.48 = { by axiom 4 (f03) }
% 8.62/1.48 rd(mult(X, Y), ld(X, Z))
% 8.62/1.48
% 8.62/1.48 Lemma 9: rd(rd(mult(X, Y), ld(X, Z)), X) = mult(X, rd(Y, Z)).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(rd(mult(X, Y), ld(X, Z)), X)
% 8.62/1.48 = { by lemma 8 R->L }
% 8.62/1.48 rd(mult(mult(X, rd(Y, Z)), X), X)
% 8.62/1.48 = { by axiom 2 (f04) }
% 8.62/1.48 mult(X, rd(Y, Z))
% 8.62/1.48
% 8.62/1.48 Lemma 10: mult(X, rd(ld(X, Y), Y)) = rd(X, X).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(X, rd(ld(X, Y), Y))
% 8.62/1.48 = { by lemma 9 R->L }
% 8.62/1.48 rd(rd(mult(X, ld(X, Y)), ld(X, Y)), X)
% 8.62/1.48 = { by axiom 2 (f04) }
% 8.62/1.48 rd(X, X)
% 8.62/1.48
% 8.62/1.48 Lemma 11: rd(ld(X, Y), Y) = ld(X, rd(X, X)).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(ld(X, Y), Y)
% 8.62/1.48 = { by axiom 1 (f02) R->L }
% 8.62/1.48 ld(X, mult(X, rd(ld(X, Y), Y)))
% 8.62/1.48 = { by lemma 10 }
% 8.62/1.48 ld(X, rd(X, X))
% 8.62/1.48
% 8.62/1.48 Lemma 12: rd(X, ld(Y, X)) = Y.
% 8.62/1.48 Proof:
% 8.62/1.48 rd(X, ld(Y, X))
% 8.62/1.48 = { by axiom 3 (f01) R->L }
% 8.62/1.48 rd(mult(Y, ld(Y, X)), ld(Y, X))
% 8.62/1.48 = { by axiom 2 (f04) }
% 8.62/1.48 Y
% 8.62/1.48
% 8.62/1.48 Lemma 13: mult(X, rd(Y, mult(X, Y))) = rd(X, X).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(X, rd(Y, mult(X, Y)))
% 8.62/1.48 = { by lemma 9 R->L }
% 8.62/1.48 rd(rd(mult(X, Y), ld(X, mult(X, Y))), X)
% 8.62/1.48 = { by lemma 12 }
% 8.62/1.48 rd(X, X)
% 8.62/1.48
% 8.62/1.48 Lemma 14: rd(X, mult(Y, X)) = ld(Y, rd(Y, Y)).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(X, mult(Y, X))
% 8.62/1.48 = { by axiom 1 (f02) R->L }
% 8.62/1.48 ld(Y, mult(Y, rd(X, mult(Y, X))))
% 8.62/1.48 = { by lemma 13 }
% 8.62/1.48 ld(Y, rd(Y, Y))
% 8.62/1.48
% 8.62/1.48 Lemma 15: rd(ld(X, Y), Y) = rd(Z, mult(X, Z)).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(ld(X, Y), Y)
% 8.62/1.48 = { by lemma 11 }
% 8.62/1.48 ld(X, rd(X, X))
% 8.62/1.48 = { by lemma 14 R->L }
% 8.62/1.48 rd(Z, mult(X, Z))
% 8.62/1.48
% 8.62/1.48 Lemma 16: ld(rd(X, Y), X) = Y.
% 8.62/1.48 Proof:
% 8.62/1.48 ld(rd(X, Y), X)
% 8.62/1.48 = { by axiom 4 (f03) R->L }
% 8.62/1.48 ld(rd(X, Y), mult(rd(X, Y), Y))
% 8.62/1.48 = { by axiom 1 (f02) }
% 8.62/1.48 Y
% 8.62/1.48
% 8.62/1.48 Lemma 17: rd(X, mult(rd(Y, Z), X)) = rd(Z, Y).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(X, mult(rd(Y, Z), X))
% 8.62/1.48 = { by lemma 15 R->L }
% 8.62/1.48 rd(ld(rd(Y, Z), Y), Y)
% 8.62/1.48 = { by lemma 16 }
% 8.62/1.48 rd(Z, Y)
% 8.62/1.48
% 8.62/1.48 Lemma 18: mult(X, mult(ld(X, Y), mult(X, Z))) = mult(mult(Y, X), Z).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(X, mult(ld(X, Y), mult(X, Z)))
% 8.62/1.48 = { by axiom 5 (f05) }
% 8.62/1.48 mult(mult(mult(X, ld(X, Y)), X), Z)
% 8.62/1.48 = { by axiom 3 (f01) }
% 8.62/1.48 mult(mult(Y, X), Z)
% 8.62/1.48
% 8.62/1.48 Lemma 19: mult(mult(X, Y), ld(Y, Z)) = mult(Y, mult(ld(Y, X), Z)).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(mult(X, Y), ld(Y, Z))
% 8.62/1.48 = { by lemma 18 R->L }
% 8.62/1.48 mult(Y, mult(ld(Y, X), mult(Y, ld(Y, Z))))
% 8.62/1.48 = { by axiom 3 (f01) }
% 8.62/1.48 mult(Y, mult(ld(Y, X), Z))
% 8.62/1.48
% 8.62/1.48 Lemma 20: ld(rd(X, mult(Y, X)), Z) = mult(Y, Z).
% 8.62/1.48 Proof:
% 8.62/1.48 ld(rd(X, mult(Y, X)), Z)
% 8.62/1.48 = { by axiom 1 (f02) R->L }
% 8.62/1.48 ld(ld(Y, mult(Y, rd(X, mult(Y, X)))), Z)
% 8.62/1.48 = { by lemma 13 }
% 8.62/1.48 ld(ld(Y, rd(Y, Y)), Z)
% 8.62/1.48 = { by lemma 13 R->L }
% 8.62/1.48 ld(ld(Y, mult(Y, rd(Z, mult(Y, Z)))), Z)
% 8.62/1.48 = { by axiom 1 (f02) }
% 8.62/1.48 ld(rd(Z, mult(Y, Z)), Z)
% 8.62/1.48 = { by lemma 16 }
% 8.62/1.48 mult(Y, Z)
% 8.62/1.48
% 8.62/1.48 Lemma 21: mult(ld(X, Y), mult(X, Z)) = ld(X, mult(mult(Y, X), Z)).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(ld(X, Y), mult(X, Z))
% 8.62/1.48 = { by axiom 1 (f02) R->L }
% 8.62/1.48 ld(X, mult(X, mult(ld(X, Y), mult(X, Z))))
% 8.62/1.48 = { by lemma 18 }
% 8.62/1.48 ld(X, mult(mult(Y, X), Z))
% 8.62/1.48
% 8.62/1.48 Lemma 22: mult(X, mult(ld(X, rd(Y, X)), Z)) = mult(Y, ld(X, Z)).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(X, mult(ld(X, rd(Y, X)), Z))
% 8.62/1.48 = { by lemma 19 R->L }
% 8.62/1.48 mult(mult(rd(Y, X), X), ld(X, Z))
% 8.62/1.48 = { by axiom 4 (f03) }
% 8.62/1.48 mult(Y, ld(X, Z))
% 8.62/1.48
% 8.62/1.48 Lemma 23: mult(ld(X, rd(Y, X)), Z) = ld(X, mult(Y, ld(X, Z))).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(ld(X, rd(Y, X)), Z)
% 8.62/1.48 = { by axiom 1 (f02) R->L }
% 8.62/1.48 ld(X, mult(X, mult(ld(X, rd(Y, X)), Z)))
% 8.62/1.48 = { by lemma 22 }
% 8.62/1.48 ld(X, mult(Y, ld(X, Z)))
% 8.62/1.48
% 8.62/1.48 Lemma 24: mult(rd(X, mult(Y, X)), Z) = ld(Y, Z).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(rd(X, mult(Y, X)), Z)
% 8.62/1.48 = { by lemma 15 R->L }
% 8.62/1.48 mult(rd(ld(Y, W), W), Z)
% 8.62/1.48 = { by lemma 11 }
% 8.62/1.48 mult(ld(Y, rd(Y, Y)), Z)
% 8.62/1.48 = { by lemma 23 }
% 8.62/1.48 ld(Y, mult(Y, ld(Y, Z)))
% 8.62/1.48 = { by axiom 1 (f02) }
% 8.62/1.48 ld(Y, Z)
% 8.62/1.48
% 8.62/1.48 Lemma 25: mult(rd(X, Y), Z) = ld(rd(Y, X), Z).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(rd(X, Y), Z)
% 8.62/1.48 = { by lemma 17 R->L }
% 8.62/1.48 mult(rd(W, mult(rd(Y, X), W)), Z)
% 8.62/1.48 = { by lemma 24 }
% 8.62/1.48 ld(rd(Y, X), Z)
% 8.62/1.48
% 8.62/1.48 Lemma 26: mult(ld(X, X), Y) = ld(rd(X, X), Y).
% 8.62/1.48 Proof:
% 8.62/1.48 mult(ld(X, X), Y)
% 8.62/1.48 = { by axiom 1 (f02) R->L }
% 8.62/1.48 ld(X, mult(X, mult(ld(X, X), Y)))
% 8.62/1.48 = { by lemma 19 R->L }
% 8.62/1.48 ld(X, mult(mult(X, X), ld(X, Y)))
% 8.62/1.48 = { by lemma 20 R->L }
% 8.62/1.48 ld(X, mult(ld(rd(Z, mult(X, Z)), X), ld(X, Y)))
% 8.62/1.48 = { by lemma 15 R->L }
% 8.62/1.48 ld(X, mult(ld(rd(ld(X, Y), Y), X), ld(X, Y)))
% 8.62/1.48 = { by axiom 4 (f03) R->L }
% 8.62/1.48 ld(X, mult(ld(rd(ld(X, Y), Y), X), mult(rd(ld(X, Y), Y), Y)))
% 8.62/1.48 = { by lemma 21 }
% 8.62/1.48 ld(X, ld(rd(ld(X, Y), Y), mult(mult(X, rd(ld(X, Y), Y)), Y)))
% 8.62/1.48 = { by lemma 10 }
% 8.62/1.48 ld(X, ld(rd(ld(X, Y), Y), mult(rd(X, X), Y)))
% 8.62/1.48 = { by axiom 1 (f02) R->L }
% 8.62/1.48 ld(X, ld(ld(X, mult(X, rd(ld(X, Y), Y))), mult(rd(X, X), Y)))
% 8.62/1.48 = { by lemma 10 }
% 8.62/1.48 ld(X, ld(ld(X, rd(X, X)), mult(rd(X, X), Y)))
% 8.62/1.48 = { by lemma 10 R->L }
% 8.62/1.48 ld(X, ld(ld(X, mult(X, rd(ld(X, W), W))), mult(rd(X, X), Y)))
% 8.62/1.48 = { by axiom 1 (f02) }
% 8.62/1.48 ld(X, ld(rd(ld(X, W), W), mult(rd(X, X), Y)))
% 8.62/1.48 = { by lemma 15 }
% 8.62/1.48 ld(X, ld(rd(V, mult(X, V)), mult(rd(X, X), Y)))
% 8.62/1.48 = { by lemma 20 }
% 8.62/1.48 ld(X, mult(X, mult(rd(X, X), Y)))
% 8.62/1.48 = { by lemma 25 }
% 8.62/1.48 ld(X, mult(X, ld(rd(X, X), Y)))
% 8.62/1.48 = { by axiom 1 (f02) }
% 8.62/1.48 ld(rd(X, X), Y)
% 8.62/1.48
% 8.62/1.48 Lemma 27: rd(X, X) = ld(X, X).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(X, X)
% 8.62/1.48 = { by lemma 17 R->L }
% 8.62/1.48 rd(Y, mult(rd(X, X), Y))
% 8.62/1.48 = { by lemma 15 R->L }
% 8.62/1.48 rd(ld(rd(X, X), Z), Z)
% 8.62/1.48 = { by lemma 26 R->L }
% 8.62/1.48 rd(mult(ld(X, X), Z), Z)
% 8.62/1.48 = { by axiom 2 (f04) }
% 8.62/1.48 ld(X, X)
% 8.62/1.48
% 8.62/1.48 Lemma 28: ld(ld(X, X), X) = X.
% 8.62/1.48 Proof:
% 8.62/1.48 ld(ld(X, X), X)
% 8.62/1.48 = { by lemma 27 R->L }
% 8.62/1.48 ld(rd(X, X), X)
% 8.62/1.48 = { by lemma 16 }
% 8.62/1.48 X
% 8.62/1.48
% 8.62/1.48 Lemma 29: ld(ld(X, X), mult(X, Y)) = ld(X, mult(mult(X, X), Y)).
% 8.62/1.48 Proof:
% 8.62/1.48 ld(ld(X, X), mult(X, Y))
% 8.62/1.48 = { by lemma 20 R->L }
% 8.62/1.48 ld(ld(X, X), ld(rd(Z, mult(X, Z)), Y))
% 8.62/1.48 = { by lemma 27 R->L }
% 8.62/1.48 ld(rd(X, X), ld(rd(Z, mult(X, Z)), Y))
% 8.62/1.48 = { by lemma 26 R->L }
% 8.62/1.48 mult(ld(X, X), ld(rd(Z, mult(X, Z)), Y))
% 8.62/1.48 = { by lemma 22 R->L }
% 8.62/1.48 mult(rd(Z, mult(X, Z)), mult(ld(rd(Z, mult(X, Z)), rd(ld(X, X), rd(Z, mult(X, Z)))), Y))
% 8.62/1.48 = { by lemma 15 R->L }
% 8.62/1.48 mult(rd(Z, mult(X, Z)), mult(ld(rd(Z, mult(X, Z)), rd(ld(X, X), rd(ld(X, W), W))), Y))
% 8.62/1.48 = { by lemma 27 R->L }
% 8.62/1.48 mult(rd(Z, mult(X, Z)), mult(ld(rd(Z, mult(X, Z)), rd(rd(X, X), rd(ld(X, W), W))), Y))
% 8.62/1.48 = { by lemma 10 R->L }
% 8.62/1.48 mult(rd(Z, mult(X, Z)), mult(ld(rd(Z, mult(X, Z)), rd(mult(X, rd(ld(X, W), W)), rd(ld(X, W), W))), Y))
% 8.62/1.48 = { by axiom 2 (f04) }
% 8.62/1.48 mult(rd(Z, mult(X, Z)), mult(ld(rd(Z, mult(X, Z)), X), Y))
% 8.62/1.48 = { by lemma 24 }
% 8.62/1.48 ld(X, mult(ld(rd(Z, mult(X, Z)), X), Y))
% 8.62/1.48 = { by lemma 20 }
% 8.62/1.48 ld(X, mult(mult(X, X), Y))
% 8.62/1.48
% 8.62/1.48 Lemma 30: rd(X, mult(Y, X)) = ld(Y, ld(Y, Y)).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(X, mult(Y, X))
% 8.62/1.48 = { by lemma 14 }
% 8.62/1.48 ld(Y, rd(Y, Y))
% 8.62/1.48 = { by lemma 27 }
% 8.62/1.48 ld(Y, ld(Y, Y))
% 8.62/1.48
% 8.62/1.48 Lemma 31: rd(ld(X, mult(mult(Y, X), Z)), mult(X, Z)) = ld(X, Y).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(ld(X, mult(mult(Y, X), Z)), mult(X, Z))
% 8.62/1.48 = { by lemma 21 R->L }
% 8.62/1.48 rd(mult(ld(X, Y), mult(X, Z)), mult(X, Z))
% 8.62/1.48 = { by axiom 2 (f04) }
% 8.62/1.48 ld(X, Y)
% 8.62/1.48
% 8.62/1.48 Lemma 32: rd(ld(X, mult(Y, Z)), mult(X, Z)) = ld(X, rd(Y, X)).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(ld(X, mult(Y, Z)), mult(X, Z))
% 8.62/1.48 = { by axiom 4 (f03) R->L }
% 8.62/1.48 rd(ld(X, mult(mult(rd(Y, X), X), Z)), mult(X, Z))
% 8.62/1.48 = { by lemma 31 }
% 8.62/1.48 ld(X, rd(Y, X))
% 8.62/1.48
% 8.62/1.48 Lemma 33: rd(ld(X, Y), mult(X, Z)) = ld(X, rd(rd(Y, Z), X)).
% 8.62/1.48 Proof:
% 8.62/1.48 rd(ld(X, Y), mult(X, Z))
% 8.62/1.48 = { by axiom 4 (f03) R->L }
% 8.62/1.48 rd(ld(X, mult(rd(Y, Z), Z)), mult(X, Z))
% 8.62/1.48 = { by lemma 32 }
% 8.62/1.48 ld(X, rd(rd(Y, Z), X))
% 8.62/1.49
% 8.62/1.49 Lemma 34: mult(mult(X, rd(X, Y)), Y) = mult(rd(X, Y), mult(Y, X)).
% 8.62/1.49 Proof:
% 8.62/1.49 mult(mult(X, rd(X, Y)), Y)
% 8.62/1.49 = { by axiom 4 (f03) R->L }
% 8.62/1.49 mult(mult(mult(rd(X, Y), Y), rd(X, Y)), Y)
% 8.62/1.49 = { by axiom 5 (f05) R->L }
% 8.62/1.49 mult(rd(X, Y), mult(Y, mult(rd(X, Y), Y)))
% 8.62/1.49 = { by axiom 4 (f03) }
% 8.62/1.49 mult(rd(X, Y), mult(Y, X))
% 8.62/1.49
% 8.62/1.49 Lemma 35: mult(rd(X, X), mult(X, X)) = rd(mult(X, X), ld(X, X)).
% 8.62/1.49 Proof:
% 8.62/1.49 mult(rd(X, X), mult(X, X))
% 8.62/1.49 = { by lemma 34 R->L }
% 8.62/1.49 mult(mult(X, rd(X, X)), X)
% 8.62/1.49 = { by lemma 8 }
% 8.62/1.49 rd(mult(X, X), ld(X, X))
% 8.62/1.49
% 8.62/1.49 Lemma 36: rd(ld(X, X), mult(X, X)) = rd(Y, mult(mult(X, X), Y)).
% 8.62/1.49 Proof:
% 8.62/1.49 rd(ld(X, X), mult(X, X))
% 8.62/1.49 = { by lemma 12 R->L }
% 8.62/1.49 rd(Y, ld(rd(ld(X, X), mult(X, X)), Y))
% 8.62/1.49 = { by lemma 25 R->L }
% 8.62/1.49 rd(Y, mult(rd(mult(X, X), ld(X, X)), Y))
% 8.62/1.49 = { by lemma 35 R->L }
% 8.62/1.49 rd(Y, mult(mult(rd(X, X), mult(X, X)), Y))
% 8.62/1.49 = { by lemma 34 R->L }
% 8.62/1.49 rd(Y, mult(mult(mult(X, rd(X, X)), X), Y))
% 8.62/1.49 = { by axiom 5 (f05) R->L }
% 8.62/1.49 rd(Y, mult(X, mult(rd(X, X), mult(X, Y))))
% 8.62/1.49 = { by lemma 25 }
% 8.62/1.49 rd(Y, mult(X, ld(rd(X, X), mult(X, Y))))
% 8.62/1.49 = { by lemma 27 }
% 8.62/1.49 rd(Y, mult(X, ld(ld(X, X), mult(X, Y))))
% 8.62/1.49 = { by lemma 29 }
% 8.62/1.49 rd(Y, mult(X, ld(X, mult(mult(X, X), Y))))
% 8.62/1.49 = { by axiom 3 (f01) }
% 8.62/1.49 rd(Y, mult(mult(X, X), Y))
% 8.62/1.49
% 8.62/1.49 Lemma 37: ld(mult(X, X), mult(X, X)) = ld(X, X).
% 8.62/1.49 Proof:
% 8.62/1.49 ld(mult(X, X), mult(X, X))
% 8.62/1.49 = { by lemma 24 R->L }
% 8.62/1.49 mult(rd(Y, mult(mult(X, X), Y)), mult(X, X))
% 8.62/1.49 = { by lemma 36 R->L }
% 8.62/1.49 mult(rd(ld(X, X), mult(X, X)), mult(X, X))
% 8.62/1.49 = { by axiom 4 (f03) }
% 8.62/1.49 ld(X, X)
% 8.62/1.49
% 8.62/1.49 Lemma 38: mult(X, ld(mult(Y, X), Z)) = ld(ld(X, Y), ld(X, Z)).
% 8.62/1.49 Proof:
% 8.62/1.49 mult(X, ld(mult(Y, X), Z))
% 8.62/1.49 = { by axiom 1 (f02) R->L }
% 8.62/1.49 ld(ld(X, Y), mult(ld(X, Y), mult(X, ld(mult(Y, X), Z))))
% 8.62/1.49 = { by lemma 21 }
% 8.62/1.49 ld(ld(X, Y), ld(X, mult(mult(Y, X), ld(mult(Y, X), Z))))
% 8.62/1.49 = { by axiom 3 (f01) }
% 8.62/1.49 ld(ld(X, Y), ld(X, Z))
% 8.62/1.49
% 8.62/1.49 Lemma 39: mult(ld(X, Y), mult(ld(ld(X, Y), X), Z)) = mult(Y, ld(ld(X, Y), Z)).
% 8.62/1.49 Proof:
% 8.62/1.49 mult(ld(X, Y), mult(ld(ld(X, Y), X), Z))
% 8.62/1.49 = { by lemma 19 R->L }
% 8.62/1.49 mult(mult(X, ld(X, Y)), ld(ld(X, Y), Z))
% 8.62/1.49 = { by axiom 3 (f01) }
% 8.62/1.49 mult(Y, ld(ld(X, Y), Z))
% 8.62/1.49
% 8.62/1.49 Lemma 40: ld(ld(X, X), ld(X, X)) = ld(X, X).
% 8.62/1.49 Proof:
% 8.62/1.49 ld(ld(X, X), ld(X, X))
% 8.62/1.49 = { by lemma 27 R->L }
% 8.62/1.49 ld(rd(X, X), ld(X, X))
% 8.62/1.49 = { by lemma 26 R->L }
% 8.62/1.49 mult(ld(X, X), ld(X, X))
% 8.62/1.49 = { by axiom 1 (f02) R->L }
% 8.62/1.49 ld(X, mult(X, mult(ld(X, X), ld(X, X))))
% 8.62/1.49 = { by lemma 19 R->L }
% 8.62/1.49 ld(X, mult(mult(X, X), ld(X, ld(X, X))))
% 8.62/1.49 = { by lemma 29 R->L }
% 8.62/1.49 ld(ld(X, X), mult(X, ld(X, ld(X, X))))
% 8.62/1.49 = { by axiom 3 (f01) R->L }
% 8.62/1.49 ld(ld(X, X), mult(X, mult(X, ld(X, ld(X, ld(X, X))))))
% 8.62/1.49 = { by lemma 30 R->L }
% 8.62/1.49 ld(ld(X, X), mult(X, mult(X, ld(X, rd(Y, mult(X, Y))))))
% 8.62/1.49 = { by lemma 15 R->L }
% 8.62/1.49 ld(ld(X, X), mult(X, mult(X, ld(X, rd(ld(X, X), X)))))
% 8.62/1.49 = { by lemma 27 R->L }
% 8.62/1.49 ld(ld(X, X), mult(X, mult(X, ld(X, rd(rd(X, X), X)))))
% 8.62/1.49 = { by lemma 33 R->L }
% 8.62/1.49 ld(ld(X, X), mult(X, mult(X, rd(ld(X, X), mult(X, X)))))
% 8.62/1.49 = { by lemma 36 }
% 8.62/1.49 ld(ld(X, X), mult(X, mult(X, rd(Z, mult(mult(X, X), Z)))))
% 8.62/1.49 = { by lemma 14 }
% 8.62/1.49 ld(ld(X, X), mult(X, mult(X, ld(mult(X, X), rd(mult(X, X), mult(X, X))))))
% 8.62/1.49 = { by lemma 27 }
% 8.62/1.49 ld(ld(X, X), mult(X, mult(X, ld(mult(X, X), ld(mult(X, X), mult(X, X))))))
% 8.62/1.49 = { by lemma 37 }
% 8.62/1.49 ld(ld(X, X), mult(X, mult(X, ld(mult(X, X), ld(X, X)))))
% 8.62/1.49 = { by lemma 38 }
% 8.62/1.49 ld(ld(X, X), mult(X, ld(ld(X, X), ld(X, ld(X, X)))))
% 8.62/1.49 = { by lemma 39 R->L }
% 8.62/1.49 ld(ld(X, X), mult(ld(X, X), mult(ld(ld(X, X), X), ld(X, ld(X, X)))))
% 8.62/1.49 = { by axiom 1 (f02) }
% 8.62/1.49 mult(ld(ld(X, X), X), ld(X, ld(X, X)))
% 8.62/1.49 = { by lemma 28 }
% 8.62/1.49 mult(X, ld(X, ld(X, X)))
% 8.62/1.49 = { by axiom 3 (f01) }
% 8.62/1.49 ld(X, X)
% 8.62/1.49
% 8.62/1.49 Lemma 41: rd(rd(X, Y), rd(X, Y)) = ld(rd(Y, X), rd(Y, X)).
% 8.62/1.49 Proof:
% 8.62/1.49 rd(rd(X, Y), rd(X, Y))
% 8.62/1.49 = { by lemma 10 R->L }
% 8.62/1.49 mult(rd(X, Y), rd(ld(rd(X, Y), Z), Z))
% 8.62/1.49 = { by lemma 25 }
% 8.62/1.49 ld(rd(Y, X), rd(ld(rd(X, Y), Z), Z))
% 8.62/1.49 = { by lemma 15 }
% 8.62/1.49 ld(rd(Y, X), rd(W, mult(rd(X, Y), W)))
% 8.62/1.49 = { by lemma 17 }
% 8.62/1.49 ld(rd(Y, X), rd(Y, X))
% 8.62/1.49
% 8.62/1.49 Lemma 42: mult(rd(X, Y), rd(Y, X)) = rd(rd(X, Y), rd(X, Y)).
% 8.62/1.49 Proof:
% 8.62/1.49 mult(rd(X, Y), rd(Y, X))
% 8.62/1.49 = { by lemma 16 R->L }
% 8.62/1.49 mult(rd(X, Y), rd(ld(rd(X, Y), X), X))
% 8.62/1.49 = { by lemma 10 }
% 8.62/1.49 rd(rd(X, Y), rd(X, Y))
% 8.62/1.49
% 8.62/1.49 Lemma 43: mult(X, rd(ld(X, Y), Z)) = rd(rd(Y, ld(X, Z)), X).
% 8.62/1.49 Proof:
% 8.62/1.49 mult(X, rd(ld(X, Y), Z))
% 8.62/1.49 = { by lemma 9 R->L }
% 8.62/1.49 rd(rd(mult(X, ld(X, Y)), ld(X, Z)), X)
% 8.62/1.49 = { by axiom 3 (f01) }
% 8.62/1.49 rd(rd(Y, ld(X, Z)), X)
% 8.62/1.49
% 8.62/1.49 Lemma 44: ld(ld(X, rd(Y, X)), ld(X, Z)) = mult(X, ld(Y, Z)).
% 8.62/1.49 Proof:
% 8.62/1.49 ld(ld(X, rd(Y, X)), ld(X, Z))
% 8.62/1.49 = { by lemma 38 R->L }
% 8.62/1.49 mult(X, ld(mult(rd(Y, X), X), Z))
% 8.62/1.49 = { by axiom 4 (f03) }
% 8.62/1.49 mult(X, ld(Y, Z))
% 8.62/1.49
% 8.62/1.49 Lemma 45: mult(X, ld(Y, mult(X, Z))) = ld(ld(X, rd(Y, X)), Z).
% 8.62/1.49 Proof:
% 8.62/1.49 mult(X, ld(Y, mult(X, Z)))
% 8.62/1.49 = { by lemma 44 R->L }
% 8.62/1.49 ld(ld(X, rd(Y, X)), ld(X, mult(X, Z)))
% 8.62/1.49 = { by axiom 1 (f02) }
% 8.62/1.49 ld(ld(X, rd(Y, X)), Z)
% 8.62/1.49
% 8.62/1.49 Lemma 46: rd(mult(X, Y), ld(X, ld(Z, Y))) = mult(mult(X, Z), X).
% 8.62/1.49 Proof:
% 8.62/1.49 rd(mult(X, Y), ld(X, ld(Z, Y)))
% 8.62/1.49 = { by axiom 3 (f01) R->L }
% 8.62/1.49 rd(mult(X, mult(Z, ld(Z, Y))), ld(X, ld(Z, Y)))
% 8.62/1.49 = { by lemma 7 }
% 8.62/1.49 mult(mult(X, Z), X)
% 8.62/1.49
% 8.62/1.49 Lemma 47: mult(ld(rd(X, Y), Z), rd(Y, X)) = rd(Y, ld(rd(Y, X), ld(Z, X))).
% 8.62/1.49 Proof:
% 8.62/1.49 mult(ld(rd(X, Y), Z), rd(Y, X))
% 8.62/1.49 = { by lemma 25 R->L }
% 8.62/1.49 mult(mult(rd(Y, X), Z), rd(Y, X))
% 8.62/1.49 = { by lemma 46 R->L }
% 8.62/1.49 rd(mult(rd(Y, X), X), ld(rd(Y, X), ld(Z, X)))
% 8.62/1.49 = { by axiom 4 (f03) }
% 8.62/1.49 rd(Y, ld(rd(Y, X), ld(Z, X)))
% 8.62/1.49
% 8.62/1.49 Lemma 48: mult(mult(rd(X, Y), Z), rd(X, Y)) = rd(mult(rd(X, Y), mult(Z, X)), Y).
% 8.62/1.49 Proof:
% 8.62/1.49 mult(mult(rd(X, Y), Z), rd(X, Y))
% 8.62/1.49 = { by lemma 6 R->L }
% 8.62/1.49 rd(mult(rd(X, Y), mult(Z, mult(rd(X, Y), Y))), Y)
% 8.62/1.49 = { by axiom 4 (f03) }
% 8.62/1.49 rd(mult(rd(X, Y), mult(Z, X)), Y)
% 8.62/1.49
% 8.62/1.49 Lemma 49: rd(X, ld(Y, ld(Z, ld(Y, X)))) = mult(mult(Y, Z), Y).
% 8.62/1.49 Proof:
% 8.62/1.49 rd(X, ld(Y, ld(Z, ld(Y, X))))
% 8.62/1.49 = { by axiom 3 (f01) R->L }
% 8.62/1.49 rd(mult(Y, ld(Y, X)), ld(Y, ld(Z, ld(Y, X))))
% 8.62/1.49 = { by lemma 46 }
% 8.62/1.49 mult(mult(Y, Z), Y)
% 8.62/1.49
% 8.62/1.49 Lemma 50: ld(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))) = rd(mult(Y, X), ld(Y, X)).
% 8.62/1.49 Proof:
% 8.62/1.49 ld(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y)))
% 8.62/1.49 = { by lemma 27 R->L }
% 8.62/1.49 ld(ld(X, X), mult(ld(rd(X, X), Y), ld(ld(X, X), Y)))
% 8.62/1.49 = { by lemma 26 R->L }
% 8.62/1.49 ld(ld(X, X), mult(mult(ld(X, X), Y), ld(ld(X, X), Y)))
% 8.62/1.49 = { by lemma 23 R->L }
% 8.62/1.49 mult(ld(ld(X, X), rd(mult(ld(X, X), Y), ld(X, X))), Y)
% 8.62/1.49 = { by lemma 12 R->L }
% 8.62/1.49 mult(rd(Y, ld(ld(ld(X, X), rd(mult(ld(X, X), Y), ld(X, X))), Y)), Y)
% 8.62/1.49 = { by lemma 45 R->L }
% 8.62/1.49 mult(rd(Y, mult(ld(X, X), ld(mult(ld(X, X), Y), mult(ld(X, X), Y)))), Y)
% 8.62/1.49 = { by lemma 20 R->L }
% 8.62/1.49 mult(rd(Y, ld(rd(Y, mult(ld(X, X), Y)), ld(mult(ld(X, X), Y), mult(ld(X, X), Y)))), Y)
% 8.62/1.49 = { by lemma 47 R->L }
% 8.62/1.49 mult(mult(ld(rd(mult(ld(X, X), Y), Y), mult(ld(X, X), Y)), rd(Y, mult(ld(X, X), Y))), Y)
% 8.62/1.49 = { by lemma 16 }
% 8.62/1.49 mult(mult(Y, rd(Y, mult(ld(X, X), Y))), Y)
% 8.62/1.49 = { by lemma 30 }
% 8.62/1.49 mult(mult(Y, ld(ld(X, X), ld(ld(X, X), ld(X, X)))), Y)
% 8.62/1.49 = { by lemma 27 R->L }
% 8.62/1.49 mult(mult(Y, ld(ld(X, X), rd(ld(X, X), ld(X, X)))), Y)
% 8.62/1.49 = { by lemma 11 R->L }
% 8.62/1.49 mult(mult(Y, rd(ld(ld(X, X), Z), Z)), Y)
% 8.62/1.49 = { by lemma 8 }
% 8.62/1.49 rd(mult(Y, ld(ld(X, X), Z)), ld(Y, Z))
% 8.62/1.49 = { by lemma 20 R->L }
% 8.62/1.49 rd(ld(rd(W, mult(Y, W)), ld(ld(X, X), Z)), ld(Y, Z))
% 8.62/1.49 = { by lemma 15 R->L }
% 8.62/1.49 rd(ld(rd(ld(Y, Z), Z), ld(ld(X, X), Z)), ld(Y, Z))
% 8.62/1.49 = { by axiom 2 (f04) R->L }
% 8.62/1.49 rd(ld(rd(ld(Y, Z), Z), ld(rd(mult(ld(X, X), ld(Y, Z)), ld(Y, Z)), Z)), ld(Y, Z))
% 8.62/1.49 = { by axiom 4 (f03) R->L }
% 8.62/1.49 rd(ld(rd(ld(Y, Z), Z), ld(rd(mult(ld(X, X), ld(Y, Z)), ld(Y, Z)), Z)), mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z))))
% 8.62/1.49 = { by lemma 25 R->L }
% 8.62/1.49 rd(ld(rd(ld(Y, Z), Z), mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), Z)), mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z))))
% 8.62/1.49 = { by axiom 4 (f03) R->L }
% 8.62/1.49 rd(ld(rd(mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z))), Z), mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), Z)), mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z))))
% 8.62/1.50 = { by lemma 25 R->L }
% 8.62/1.50 rd(mult(rd(Z, mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z)))), mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), Z)), mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z))))
% 8.62/1.50 = { by lemma 48 R->L }
% 8.62/1.50 mult(mult(rd(Z, mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z)))), rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z)))), rd(Z, mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z)))))
% 8.62/1.50 = { by lemma 7 R->L }
% 8.62/1.50 rd(mult(rd(Z, mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z)))), mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z)))), ld(rd(Z, mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z)))), mult(ld(X, X), ld(Y, Z))))
% 8.62/1.50 = { by axiom 4 (f03) }
% 8.62/1.50 rd(Z, ld(rd(Z, mult(rd(ld(Y, Z), mult(ld(X, X), ld(Y, Z))), mult(ld(X, X), ld(Y, Z)))), mult(ld(X, X), ld(Y, Z))))
% 8.62/1.50 = { by axiom 4 (f03) }
% 8.62/1.50 rd(Z, ld(rd(Z, ld(Y, Z)), mult(ld(X, X), ld(Y, Z))))
% 8.62/1.50 = { by lemma 12 }
% 8.62/1.50 rd(Z, ld(Y, mult(ld(X, X), ld(Y, Z))))
% 8.62/1.50 = { by lemma 26 }
% 8.62/1.50 rd(Z, ld(Y, ld(rd(X, X), ld(Y, Z))))
% 8.62/1.50 = { by lemma 49 }
% 8.62/1.50 mult(mult(Y, rd(X, X)), Y)
% 8.62/1.50 = { by lemma 8 }
% 8.62/1.50 rd(mult(Y, X), ld(Y, X))
% 8.62/1.50
% 8.62/1.50 Lemma 51: rd(mult(X, X), ld(X, X)) = mult(X, X).
% 8.62/1.50 Proof:
% 8.62/1.50 rd(mult(X, X), ld(X, X))
% 8.62/1.50 = { by lemma 37 R->L }
% 8.62/1.50 rd(mult(X, X), ld(mult(X, X), mult(X, X)))
% 8.62/1.50 = { by lemma 12 }
% 8.62/1.50 mult(X, X)
% 8.62/1.50
% 8.62/1.50 Lemma 52: rd(rd(mult(X, X), ld(X, X)), mult(X, X)) = ld(X, X).
% 8.62/1.50 Proof:
% 8.62/1.50 rd(rd(mult(X, X), ld(X, X)), mult(X, X))
% 8.62/1.50 = { by lemma 35 R->L }
% 8.62/1.50 rd(mult(rd(X, X), mult(X, X)), mult(X, X))
% 8.62/1.50 = { by axiom 2 (f04) }
% 8.62/1.50 rd(X, X)
% 8.62/1.50 = { by lemma 27 }
% 8.62/1.50 ld(X, X)
% 8.62/1.50
% 8.62/1.50 Lemma 53: ld(ld(X, X), rd(ld(X, X), Y)) = rd(Z, mult(Y, Z)).
% 8.62/1.50 Proof:
% 8.62/1.50 ld(ld(X, X), rd(ld(X, X), Y))
% 8.62/1.50 = { by axiom 1 (f02) R->L }
% 8.62/1.50 ld(Y, mult(Y, ld(ld(X, X), rd(ld(X, X), Y))))
% 8.62/1.50 = { by lemma 27 R->L }
% 8.62/1.50 ld(Y, mult(Y, ld(rd(X, X), rd(ld(X, X), Y))))
% 8.62/1.50 = { by lemma 27 R->L }
% 8.62/1.50 ld(Y, mult(Y, ld(rd(X, X), rd(rd(X, X), Y))))
% 8.62/1.50 = { by lemma 44 R->L }
% 8.62/1.50 ld(Y, ld(ld(Y, rd(rd(X, X), Y)), ld(Y, rd(rd(X, X), Y))))
% 8.62/1.50 = { by lemma 33 R->L }
% 8.62/1.50 ld(Y, ld(rd(ld(Y, X), mult(Y, X)), ld(Y, rd(rd(X, X), Y))))
% 8.62/1.50 = { by lemma 33 R->L }
% 8.62/1.50 ld(Y, ld(rd(ld(Y, X), mult(Y, X)), rd(ld(Y, X), mult(Y, X))))
% 8.62/1.50 = { by lemma 41 R->L }
% 8.62/1.50 ld(Y, rd(rd(mult(Y, X), ld(Y, X)), rd(mult(Y, X), ld(Y, X))))
% 8.62/1.50 = { by lemma 50 R->L }
% 8.62/1.50 ld(Y, rd(ld(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), rd(mult(Y, X), ld(Y, X))))
% 8.62/1.50 = { by lemma 27 R->L }
% 8.62/1.50 ld(Y, rd(ld(rd(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), rd(mult(Y, X), ld(Y, X))))
% 8.62/1.50 = { by lemma 26 R->L }
% 8.62/1.50 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), rd(mult(Y, X), ld(Y, X))))
% 8.99/1.50 = { by lemma 50 R->L }
% 8.99/1.50 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y)))))
% 8.99/1.50 = { by lemma 51 R->L }
% 8.99/1.50 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(ld(X, X), rd(mult(ld(ld(X, X), Y), ld(ld(X, X), Y)), ld(ld(ld(X, X), Y), ld(ld(X, X), Y))))))
% 8.99/1.50 = { by lemma 35 R->L }
% 8.99/1.50 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(ld(X, X), mult(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))))))
% 8.99/1.50 = { by axiom 1 (f02) R->L }
% 8.99/1.50 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(ld(X, X), mult(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), ld(ld(X, X), mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))))))))
% 8.99/1.50 = { by lemma 25 }
% 8.99/1.50 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(ld(X, X), ld(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), ld(ld(X, X), mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))))))))
% 8.99/1.50 = { by lemma 49 }
% 8.99/1.50 ld(Y, mult(mult(ld(X, X), rd(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(X, X)))
% 8.99/1.50 = { by lemma 8 }
% 8.99/1.50 ld(Y, rd(mult(ld(X, X), ld(ld(X, X), Y)), ld(ld(X, X), ld(ld(X, X), Y))))
% 8.99/1.50 = { by lemma 26 }
% 8.99/1.50 ld(Y, rd(ld(rd(X, X), ld(ld(X, X), Y)), ld(ld(X, X), ld(ld(X, X), Y))))
% 8.99/1.50 = { by lemma 27 }
% 8.99/1.50 ld(Y, rd(ld(ld(X, X), ld(ld(X, X), Y)), ld(ld(X, X), ld(ld(X, X), Y))))
% 8.99/1.50 = { by lemma 37 R->L }
% 8.99/1.50 ld(Y, rd(ld(ld(X, X), ld(ld(X, X), Y)), ld(ld(mult(X, X), mult(X, X)), ld(ld(X, X), Y))))
% 8.99/1.50 = { by lemma 27 R->L }
% 8.99/1.50 ld(Y, rd(ld(ld(X, X), ld(ld(X, X), Y)), ld(rd(mult(X, X), mult(X, X)), ld(ld(X, X), Y))))
% 8.99/1.50 = { by lemma 51 R->L }
% 8.99/1.50 ld(Y, rd(ld(ld(X, X), ld(ld(X, X), Y)), ld(rd(mult(X, X), rd(mult(X, X), ld(X, X))), ld(ld(X, X), Y))))
% 8.99/1.50 = { by lemma 52 R->L }
% 8.99/1.50 ld(Y, rd(ld(rd(rd(mult(X, X), ld(X, X)), mult(X, X)), ld(ld(X, X), Y)), ld(rd(mult(X, X), rd(mult(X, X), ld(X, X))), ld(ld(X, X), Y))))
% 8.99/1.50 = { by lemma 25 R->L }
% 8.99/1.50 ld(Y, rd(mult(rd(mult(X, X), rd(mult(X, X), ld(X, X))), ld(ld(X, X), Y)), ld(rd(mult(X, X), rd(mult(X, X), ld(X, X))), ld(ld(X, X), Y))))
% 8.99/1.50 = { by lemma 8 R->L }
% 8.99/1.50 ld(Y, mult(mult(rd(mult(X, X), rd(mult(X, X), ld(X, X))), rd(ld(ld(X, X), Y), ld(ld(X, X), Y))), rd(mult(X, X), rd(mult(X, X), ld(X, X)))))
% 8.99/1.50 = { by lemma 48 }
% 8.99/1.50 ld(Y, rd(mult(rd(mult(X, X), rd(mult(X, X), ld(X, X))), mult(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), mult(X, X))), rd(mult(X, X), ld(X, X))))
% 8.99/1.50 = { by lemma 25 }
% 8.99/1.50 ld(Y, rd(ld(rd(rd(mult(X, X), ld(X, X)), mult(X, X)), mult(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), mult(X, X))), rd(mult(X, X), ld(X, X))))
% 8.99/1.50 = { by axiom 4 (f03) R->L }
% 8.99/1.50 ld(Y, rd(ld(rd(rd(mult(X, X), ld(X, X)), mult(X, X)), mult(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), mult(X, X))), mult(rd(rd(mult(X, X), ld(X, X)), mult(X, X)), mult(X, X))))
% 8.99/1.50 = { by lemma 32 }
% 8.99/1.50 ld(Y, ld(rd(rd(mult(X, X), ld(X, X)), mult(X, X)), rd(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), rd(rd(mult(X, X), ld(X, X)), mult(X, X)))))
% 8.99/1.50 = { by lemma 52 }
% 8.99/1.50 ld(Y, ld(ld(X, X), rd(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), rd(rd(mult(X, X), ld(X, X)), mult(X, X)))))
% 8.99/1.50 = { by lemma 52 }
% 8.99/1.50 ld(Y, ld(ld(X, X), rd(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), ld(X, X))))
% 8.99/1.50 = { by lemma 43 R->L }
% 8.99/1.50 ld(Y, ld(ld(X, X), mult(ld(X, X), rd(ld(ld(X, X), ld(ld(X, X), Y)), Y))))
% 8.99/1.50 = { by axiom 1 (f02) }
% 8.99/1.50 ld(Y, rd(ld(ld(X, X), ld(ld(X, X), Y)), Y))
% 8.99/1.50 = { by lemma 40 R->L }
% 8.99/1.50 ld(Y, rd(ld(ld(X, X), ld(ld(ld(X, X), ld(X, X)), Y)), Y))
% 8.99/1.50 = { by lemma 27 R->L }
% 8.99/1.50 ld(Y, rd(ld(rd(X, X), ld(ld(ld(X, X), ld(X, X)), Y)), Y))
% 8.99/1.50 = { by lemma 26 R->L }
% 8.99/1.50 ld(Y, rd(mult(ld(X, X), ld(ld(ld(X, X), ld(X, X)), Y)), Y))
% 8.99/1.50 = { by lemma 39 R->L }
% 8.99/1.50 ld(Y, rd(mult(ld(ld(X, X), ld(X, X)), mult(ld(ld(ld(X, X), ld(X, X)), ld(X, X)), Y)), Y))
% 8.99/1.50 = { by lemma 40 }
% 8.99/1.50 ld(Y, rd(mult(ld(X, X), mult(ld(ld(ld(X, X), ld(X, X)), ld(X, X)), Y)), Y))
% 8.99/1.50 = { by lemma 26 }
% 8.99/1.50 ld(Y, rd(ld(rd(X, X), mult(ld(ld(ld(X, X), ld(X, X)), ld(X, X)), Y)), Y))
% 8.99/1.50 = { by lemma 27 }
% 8.99/1.50 ld(Y, rd(ld(ld(X, X), mult(ld(ld(ld(X, X), ld(X, X)), ld(X, X)), Y)), Y))
% 8.99/1.50 = { by lemma 28 }
% 8.99/1.50 ld(Y, rd(ld(ld(X, X), mult(ld(X, X), Y)), Y))
% 8.99/1.50 = { by axiom 1 (f02) }
% 8.99/1.50 ld(Y, rd(Y, Y))
% 8.99/1.50 = { by lemma 11 R->L }
% 8.99/1.50 rd(ld(Y, W), W)
% 8.99/1.50 = { by lemma 15 }
% 8.99/1.50 rd(Z, mult(Y, Z))
% 8.99/1.50
% 8.99/1.50 Lemma 54: rd(rd(ld(X, X), Y), ld(X, X)) = rd(Z, mult(ld(ld(X, X), Y), Z)).
% 8.99/1.50 Proof:
% 8.99/1.50 rd(rd(ld(X, X), Y), ld(X, X))
% 8.99/1.50 = { by axiom 1 (f02) R->L }
% 8.99/1.50 rd(rd(ld(X, X), ld(ld(X, X), mult(ld(X, X), Y))), ld(X, X))
% 8.99/1.50 = { by lemma 40 R->L }
% 8.99/1.50 rd(rd(ld(X, X), ld(ld(ld(X, X), ld(X, X)), mult(ld(X, X), Y))), ld(X, X))
% 8.99/1.50 = { by lemma 40 R->L }
% 8.99/1.50 rd(rd(ld(X, X), ld(ld(ld(X, X), ld(X, X)), mult(ld(X, X), Y))), ld(ld(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 27 R->L }
% 8.99/1.50 rd(rd(ld(X, X), ld(ld(rd(X, X), ld(X, X)), mult(ld(X, X), Y))), ld(ld(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 27 R->L }
% 8.99/1.50 rd(rd(ld(X, X), ld(ld(rd(X, X), ld(X, X)), mult(ld(X, X), Y))), ld(rd(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 26 R->L }
% 8.99/1.50 rd(rd(ld(X, X), ld(mult(ld(X, X), ld(X, X)), mult(ld(X, X), Y))), ld(rd(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 26 R->L }
% 8.99/1.50 rd(rd(ld(X, X), ld(mult(ld(X, X), ld(X, X)), mult(ld(X, X), Y))), mult(ld(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 40 R->L }
% 8.99/1.50 rd(rd(ld(ld(X, X), ld(X, X)), ld(mult(ld(X, X), ld(X, X)), mult(ld(X, X), Y))), mult(ld(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 27 R->L }
% 8.99/1.50 rd(rd(ld(rd(X, X), ld(X, X)), ld(mult(ld(X, X), ld(X, X)), mult(ld(X, X), Y))), mult(ld(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 27 R->L }
% 8.99/1.50 rd(rd(ld(rd(X, X), rd(X, X)), ld(mult(ld(X, X), ld(X, X)), mult(ld(X, X), Y))), mult(ld(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 27 R->L }
% 8.99/1.50 rd(rd(ld(rd(X, X), rd(X, X)), ld(mult(rd(X, X), ld(X, X)), mult(ld(X, X), Y))), mult(ld(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 27 R->L }
% 8.99/1.50 rd(rd(ld(rd(X, X), rd(X, X)), ld(mult(rd(X, X), rd(X, X)), mult(ld(X, X), Y))), mult(ld(X, X), ld(X, X)))
% 8.99/1.50 = { by lemma 27 R->L }
% 8.99/1.50 rd(rd(ld(rd(X, X), rd(X, X)), ld(mult(rd(X, X), rd(X, X)), mult(ld(X, X), Y))), mult(rd(X, X), ld(X, X)))
% 8.99/1.51 = { by lemma 27 R->L }
% 8.99/1.51 rd(rd(ld(rd(X, X), rd(X, X)), ld(mult(rd(X, X), rd(X, X)), mult(ld(X, X), Y))), mult(rd(X, X), rd(X, X)))
% 8.99/1.51 = { by lemma 41 R->L }
% 8.99/1.51 rd(rd(rd(rd(X, X), rd(X, X)), ld(mult(rd(X, X), rd(X, X)), mult(ld(X, X), Y))), mult(rd(X, X), rd(X, X)))
% 8.99/1.51 = { by lemma 42 R->L }
% 8.99/1.51 rd(rd(mult(rd(X, X), rd(X, X)), ld(mult(rd(X, X), rd(X, X)), mult(ld(X, X), Y))), mult(rd(X, X), rd(X, X)))
% 8.99/1.51 = { by lemma 43 R->L }
% 8.99/1.51 mult(mult(rd(X, X), rd(X, X)), rd(ld(mult(rd(X, X), rd(X, X)), mult(rd(X, X), rd(X, X))), mult(ld(X, X), Y)))
% 8.99/1.51 = { by lemma 37 }
% 8.99/1.51 mult(mult(rd(X, X), rd(X, X)), rd(ld(rd(X, X), rd(X, X)), mult(ld(X, X), Y)))
% 8.99/1.51 = { by lemma 42 }
% 8.99/1.51 mult(rd(rd(X, X), rd(X, X)), rd(ld(rd(X, X), rd(X, X)), mult(ld(X, X), Y)))
% 8.99/1.51 = { by lemma 25 }
% 8.99/1.51 ld(rd(rd(X, X), rd(X, X)), rd(ld(rd(X, X), rd(X, X)), mult(ld(X, X), Y)))
% 8.99/1.51 = { by lemma 41 }
% 8.99/1.51 ld(ld(rd(X, X), rd(X, X)), rd(ld(rd(X, X), rd(X, X)), mult(ld(X, X), Y)))
% 8.99/1.51 = { by lemma 53 }
% 8.99/1.51 rd(Z, mult(mult(ld(X, X), Y), Z))
% 8.99/1.51 = { by lemma 26 }
% 8.99/1.51 rd(Z, mult(ld(rd(X, X), Y), Z))
% 8.99/1.51 = { by lemma 27 }
% 8.99/1.51 rd(Z, mult(ld(ld(X, X), Y), Z))
% 8.99/1.51
% 8.99/1.51 Lemma 55: ld(ld(X, X), Y) = Y.
% 8.99/1.51 Proof:
% 8.99/1.51 ld(ld(X, X), Y)
% 8.99/1.51 = { by axiom 2 (f04) R->L }
% 8.99/1.51 rd(mult(ld(ld(X, X), Y), Z), Z)
% 8.99/1.51 = { by lemma 17 R->L }
% 8.99/1.51 rd(W, mult(rd(Z, mult(ld(ld(X, X), Y), Z)), W))
% 8.99/1.51 = { by lemma 54 R->L }
% 8.99/1.51 rd(W, mult(rd(rd(ld(X, X), Y), ld(X, X)), W))
% 8.99/1.51 = { by lemma 53 R->L }
% 8.99/1.51 ld(ld(X, X), rd(ld(X, X), rd(rd(ld(X, X), Y), ld(X, X))))
% 8.99/1.51 = { by axiom 4 (f03) R->L }
% 8.99/1.51 ld(ld(X, X), mult(rd(rd(ld(X, X), rd(rd(ld(X, X), Y), ld(X, X))), ld(X, X)), ld(X, X)))
% 8.99/1.51 = { by lemma 54 }
% 8.99/1.51 ld(ld(X, X), mult(rd(V, mult(ld(ld(X, X), rd(rd(ld(X, X), Y), ld(X, X))), V)), ld(X, X)))
% 8.99/1.51 = { by lemma 24 }
% 8.99/1.51 ld(ld(X, X), ld(ld(ld(X, X), rd(rd(ld(X, X), Y), ld(X, X))), ld(X, X)))
% 8.99/1.51 = { by lemma 24 R->L }
% 8.99/1.51 mult(rd(U, mult(ld(X, X), U)), ld(ld(ld(X, X), rd(rd(ld(X, X), Y), ld(X, X))), ld(X, X)))
% 8.99/1.51 = { by lemma 12 R->L }
% 8.99/1.51 mult(rd(U, mult(ld(X, X), U)), ld(rd(U, ld(ld(ld(X, X), rd(rd(ld(X, X), Y), ld(X, X))), U)), ld(X, X)))
% 8.99/1.51 = { by lemma 45 R->L }
% 8.99/1.51 mult(rd(U, mult(ld(X, X), U)), ld(rd(U, mult(ld(X, X), ld(rd(ld(X, X), Y), mult(ld(X, X), U)))), ld(X, X)))
% 8.99/1.51 = { by lemma 20 R->L }
% 8.99/1.51 mult(rd(U, mult(ld(X, X), U)), ld(rd(U, ld(rd(U, mult(ld(X, X), U)), ld(rd(ld(X, X), Y), mult(ld(X, X), U)))), ld(X, X)))
% 8.99/1.51 = { by lemma 47 R->L }
% 8.99/1.51 mult(rd(U, mult(ld(X, X), U)), ld(mult(ld(rd(mult(ld(X, X), U), U), rd(ld(X, X), Y)), rd(U, mult(ld(X, X), U))), ld(X, X)))
% 8.99/1.51 = { by axiom 2 (f04) }
% 8.99/1.51 mult(rd(U, mult(ld(X, X), U)), ld(mult(ld(ld(X, X), rd(ld(X, X), Y)), rd(U, mult(ld(X, X), U))), ld(X, X)))
% 8.99/1.51 = { by lemma 38 }
% 8.99/1.51 ld(ld(rd(U, mult(ld(X, X), U)), ld(ld(X, X), rd(ld(X, X), Y))), ld(rd(U, mult(ld(X, X), U)), ld(X, X)))
% 8.99/1.51 = { by lemma 20 }
% 8.99/1.51 ld(mult(ld(X, X), ld(ld(X, X), rd(ld(X, X), Y))), ld(rd(U, mult(ld(X, X), U)), ld(X, X)))
% 8.99/1.51 = { by axiom 3 (f01) }
% 8.99/1.51 ld(rd(ld(X, X), Y), ld(rd(U, mult(ld(X, X), U)), ld(X, X)))
% 8.99/1.51 = { by lemma 20 }
% 8.99/1.51 ld(rd(ld(X, X), Y), mult(ld(X, X), ld(X, X)))
% 8.99/1.51 = { by lemma 26 }
% 8.99/1.51 ld(rd(ld(X, X), Y), ld(rd(X, X), ld(X, X)))
% 8.99/1.51 = { by lemma 27 }
% 8.99/1.51 ld(rd(ld(X, X), Y), ld(ld(X, X), ld(X, X)))
% 8.99/1.51 = { by lemma 40 }
% 8.99/1.51 ld(rd(ld(X, X), Y), ld(X, X))
% 8.99/1.51 = { by lemma 16 }
% 8.99/1.51 Y
% 8.99/1.51
% 8.99/1.51 Goal 1 (goals): tuple(mult(X, x1(X)), mult(x1_2(X), X)) = tuple(x1(X), x1_2(X)).
% 8.99/1.51 The goal is true when:
% 8.99/1.51 X = ld(X, X)
% 8.99/1.51
% 8.99/1.51 Proof:
% 8.99/1.51 tuple(mult(ld(X, X), x1(ld(X, X))), mult(x1_2(ld(X, X)), ld(X, X)))
% 8.99/1.51 = { by axiom 2 (f04) R->L }
% 8.99/1.51 tuple(mult(ld(X, X), x1(ld(X, X))), rd(mult(mult(x1_2(ld(X, X)), ld(X, X)), Y), Y))
% 8.99/1.51 = { by lemma 55 R->L }
% 8.99/1.51 tuple(mult(ld(X, X), x1(ld(X, X))), rd(mult(mult(x1_2(ld(X, X)), ld(X, X)), Y), ld(ld(X, X), Y)))
% 8.99/1.51 = { by lemma 27 R->L }
% 8.99/1.51 tuple(mult(ld(X, X), x1(ld(X, X))), rd(mult(mult(x1_2(ld(X, X)), ld(X, X)), Y), ld(rd(X, X), Y)))
% 8.99/1.51 = { by lemma 26 R->L }
% 8.99/1.51 tuple(mult(ld(X, X), x1(ld(X, X))), rd(mult(mult(x1_2(ld(X, X)), ld(X, X)), Y), mult(ld(X, X), Y)))
% 8.99/1.51 = { by lemma 55 R->L }
% 8.99/1.51 tuple(mult(ld(X, X), x1(ld(X, X))), rd(ld(ld(X, X), mult(mult(x1_2(ld(X, X)), ld(X, X)), Y)), mult(ld(X, X), Y)))
% 8.99/1.51 = { by lemma 31 }
% 8.99/1.51 tuple(mult(ld(X, X), x1(ld(X, X))), ld(ld(X, X), x1_2(ld(X, X))))
% 8.99/1.51 = { by lemma 55 }
% 8.99/1.51 tuple(mult(ld(X, X), x1(ld(X, X))), x1_2(ld(X, X)))
% 8.99/1.51 = { by lemma 26 }
% 8.99/1.51 tuple(ld(rd(X, X), x1(ld(X, X))), x1_2(ld(X, X)))
% 8.99/1.51 = { by lemma 27 }
% 8.99/1.51 tuple(ld(ld(X, X), x1(ld(X, X))), x1_2(ld(X, X)))
% 8.99/1.51 = { by lemma 55 }
% 8.99/1.51 tuple(x1(ld(X, X)), x1_2(ld(X, X)))
% 8.99/1.51 % SZS output end Proof
% 8.99/1.51
% 8.99/1.51 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------